# What Should SFT Learn? Conditional Information Gain for Generalizable Post-Training

arXiv: https://arxiv.org/abs/2610.12995 · HTML: https://mlsanity.com/paper/2610.12995/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 38 of its 45 references that are in the map.
- First posted on arXiv 9 Oct 2026.

## Builds on (most cited first, 30 of 38)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [PaLM: Scaling Language Modeling with Pathways](https://mlsanity.com/paper/2204.02311/) (2022)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Finetuned Language Models Are Zero-Shot Learners](https://mlsanity.com/paper/2109.01652/) (2021)
- [Scaling Instruction-Finetuned Language Models](https://mlsanity.com/paper/2210.11416/) (2022)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [LiveCodeBench: Holistic and Contamination Free Evaluation of Large Language Models for Code](https://mlsanity.com/paper/2403.07974/) (2024)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (2022)
- [Is Your Code Generated by ChatGPT Really Correct? Rigorous Evaluation of Large Language Models for Code Generation](https://mlsanity.com/paper/2305.01210/) (2023)
- [Qwen2.5-Coder Technical Report](https://mlsanity.com/paper/2409.12186/) (2024)
- [OlympiadBench: A Challenging Benchmark for Promoting AGI with Olympiad-Level Bilingual Multimodal Scientific Problems](https://mlsanity.com/paper/2402.14008/) (2024)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [SimPO: Simple Preference Optimization with a Reference-Free Reward](https://mlsanity.com/paper/2405.14734/) (2024)
- [STaR: Bootstrapping Reasoning With Reasoning](https://mlsanity.com/paper/2203.14465/) (2022)
- [On-Policy Distillation of Language Models: Learning from Self-Generated Mistakes](https://mlsanity.com/paper/2306.13649/) (2023)
- [SFT Memorizes, RL Generalizes: A Comparative Study of Foundation Model Post-training](https://mlsanity.com/paper/2501.17161/) (2025)
- [BigCodeBench: Benchmarking Code Generation with Diverse Function Calls and Complex Instructions](https://mlsanity.com/paper/2406.15877/) (2024)
- [The Entropy Mechanism of Reinforcement Learning for Reasoning Language Models](https://mlsanity.com/paper/2505.22617/) (2025)
- [Self-Distilled Reasoner: On-Policy Self-Distillation for Large Language Models](https://mlsanity.com/paper/2601.18734/) (2026)
- [DS-1000: A Natural and Reliable Benchmark for Data Science Code Generation](https://mlsanity.com/paper/2211.11501/) (2022)
- [Understanding Dataset Difficulty with V-Usable Information](https://mlsanity.com/paper/2110.08420/) (2021)
- [A Theory of Usable Information Under Computational Constraints](https://mlsanity.com/paper/2002.10689/) (2020)
- [Understanding Catastrophic Forgetting in Language Models via Implicit Inference](https://mlsanity.com/paper/2309.10105/) (2023)
- [Offline RL for Natural Language Generation with Implicit Language Q Learning](https://mlsanity.com/paper/2206.11871/) (2022)

## Abstract (arXiv)

Supervised fine-tuning (SFT) remains a widely used approach for post-training large language models, because fixed demonstrations can directly introduce desired reasoning strategies and behavioral patterns. Yet forcing a model to imitate externally provided trajectories may also interfere with behaviors already supported by pretraining, limiting generalization beyond the training distribution. Recent dynamic SFT methods mitigate this issue by reweighting token-level supervision according to model-side signals such as predictive probability, prior support, or distributional compatibility. However, model compatibility is not equivalent to learning value: a token that is weakly supported by the current model may either be an undesirable mismatch or precisely the task-specific decision that the demonstration is intended to teach. We propose CIG-SFT, an information-theoretic framework that estimates the task-conditioned information carried by each demonstrated token. Under teacher forcing, we compare the token likelihood from a frozen reference model with and without access to the task input while keeping the gold solution prefix fixed. The resulting conditional information gain quantifies how much the task itself reduces uncertainty about the demonstrated decision. We then convert this information into a KL-regularized projection of the current policy and derive an equivalent scalar-weighted SFT objective whose logit gradient exactly matches that of the projected target. Across mathematical reasoning and code-generation benchmarks, CIG-SFT consistently outperforms standard SFT and recent adaptive fine-tuning baselines, with particularly strong improvements in generalization to challenging evaluation distributions, while requiring neither additional training data nor online rollouts.